Theoretical analysis of time fractal fractional pantograph stochastic differential equations

Abstract

The fractal-fractional derivative, a significant mathematical concept that merges fractal geometry with fractional calculus, has garnered increasing attention for modeling complex systems. To the best of our knowledge, no existing work has addressed the well-posedness, regularity, and averaging principle for fractal-fractional pantograph stochastic differential equations (FFrPSDEs). In this study, we fill this gap by presenting results under the Atangana fractal-fractional derivative with the Riemann–Liouville (RL) definition and a power-law kernel. These equations capture essential features such as fractal behavior, memory effects, nonlocal dynamics, stochasticity, and time delays. We first establish the existence and uniqueness of solutions using a fixed-point approach. Next, we present results on continuous dependence and solution regularity. We also prove an averaging principle that simplifies the analysis of complex systems. Finally, illustrative examples are provided to demonstrate the applicability of the theoretical findings. © 2025 The Authors